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Theorem rngounval2 25425
Description: The value of the unit of a ring. (Contributed by FL, 12-Feb-2010.) (Revised by Mario Carneiro, 23-Dec-2013.)
Hypotheses
Ref Expression
rngunval2.1  |-  X  =  ran  ( 1st `  R
)
rngunval2.2  |-  H  =  ( 2nd `  R
)
rngunval2.3  |-  U  =  (GId `  H )
Assertion
Ref Expression
rngounval2  |-  ( R  e.  RingOps  ->  U  =  (
iota_ u  e.  X A. x  e.  X  ( ( u H x )  =  x  /\  ( x H u )  =  x ) ) )
Distinct variable groups:    u, H, x    u, X, x    u, R
Allowed substitution hints:    R( x)    U( x, u)

Proof of Theorem rngounval2
StepHypRef Expression
1 rngunval2.1 . . . 4  |-  X  =  ran  ( 1st `  R
)
2 rngunval2.2 . . . . 5  |-  H  =  ( 2nd `  R
)
3 eqid 2283 . . . . 5  |-  ( 1st `  R )  =  ( 1st `  R )
42, 3rngorn1eq 21087 . . . 4  |-  ( R  e.  RingOps  ->  ran  ( 1st `  R )  =  ran  H )
51, 4syl5eq 2327 . . 3  |-  ( R  e.  RingOps  ->  X  =  ran  H )
65raleqdv 2742 . . 3  |-  ( R  e.  RingOps  ->  ( A. x  e.  X  ( (
u H x )  =  x  /\  (
x H u )  =  x )  <->  A. x  e.  ran  H ( ( u H x )  =  x  /\  (
x H u )  =  x ) ) )
75, 6riotaeqbidv 6307 . 2  |-  ( R  e.  RingOps  ->  ( iota_ u  e.  X A. x  e.  X  ( ( u H x )  =  x  /\  ( x H u )  =  x ) )  =  ( iota_ u  e.  ran  H A. x  e.  ran  H ( ( u H x )  =  x  /\  ( x H u )  =  x ) ) )
8 rngunval2.3 . . 3  |-  U  =  (GId `  H )
9 fvex 5539 . . . . 5  |-  ( 2nd `  R )  e.  _V
102, 9eqeltri 2353 . . . 4  |-  H  e. 
_V
11 eqid 2283 . . . . 5  |-  ran  H  =  ran  H
1211gidval 20880 . . . 4  |-  ( H  e.  _V  ->  (GId `  H )  =  (
iota_ u  e.  ran  H A. x  e.  ran  H ( ( u H x )  =  x  /\  ( x H u )  =  x ) ) )
1310, 12ax-mp 8 . . 3  |-  (GId `  H )  =  (
iota_ u  e.  ran  H A. x  e.  ran  H ( ( u H x )  =  x  /\  ( x H u )  =  x ) )
148, 13eqtri 2303 . 2  |-  U  =  ( iota_ u  e.  ran  H A. x  e.  ran  H ( ( u H x )  =  x  /\  ( x H u )  =  x ) )
157, 14syl6reqr 2334 1  |-  ( R  e.  RingOps  ->  U  =  (
iota_ u  e.  X A. x  e.  X  ( ( u H x )  =  x  /\  ( x H u )  =  x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788   ran crn 4690   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   iota_crio 6297  GIdcgi 20854   RingOpscrngo 21042
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-ov 5861  df-1st 6122  df-2nd 6123  df-riota 6304  df-gid 20859  df-rngo 21043
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